Finding the Exact Distribution of (Peak) Age of Information for Queues of PH/PH/1/1 and M/PH/1/2 Type

TL;DR

This paper develops a numerical algorithm to precisely determine the distribution of Age of Information and peak AoI in bufferless and single-buffer queueing systems with probabilistic preemption and replacement, using Markov Fluid Queue theory.

Contribution

It introduces a matrix-analytical method for exact AoI distribution calculation in complex queue models with probabilistic preemption and replacement, validated against simulations and existing results.

Findings

Exact AoI distributions are obtained in matrix exponential form.

The method is validated with simulations and known closed-form results.

Performance comparison of queueing systems under different traffic conditions.

Abstract

Bufferless and single-buffer queueing systems have recently been shown to be effective in coping with escalated Age of Information (AoI) figures arising in single-source status update systems with large buffers and FCFS scheduling. In this paper, for the single-source scenario, we propose a numerical algorithm for obtaining the exact distributions of both the AoI and the peak AoI (PAoI) in (i) the bufferless $PH/PH/1/1/P(p)$ queue with probabilistic preemption with preemption probability $p$, $0 \leq p \leq 1,$ and (ii) the single buffer $M/PH/1/2/R(r)$ queue with probabilistic replacement of the packet in the queue by the new arrival with replacement probability $r$, $0 \leq r \leq 1$. The proposed exact models are based on the well-established theory of Markov Fluid Queues (MFQ) and the numerical algorithms are matrix-analytical and they rely on numerically stable and efficient vector-matrix operations. Moreover, the obtained exact distributions are in matrix exponential form, making it amenable to calculate the tail probabilities and the associated moments straightforwardly. Firstly, we validate the accuracy of the proposed method with simulations, and for sume sub-cases, with existing closed-form results. We then comparatively study the AoI performance of the queueing systems of interest under varying traffic parameters.